# Linear Inequalities  Top Download PDF

In mathematics, linear inequality is an inequality that involves a linear function. Linear inequality has one of the symbols of inequality. Shows data that is not equal in the form of a graph.

## Symbol of Inequality:

• < less than

• > greater than

• ≤ less than or equal to

• ≥ greater than or equal to

• ≠ not equal to

• = equal to

### Linear Inequation of One Variable

Let a be the non-zero real number and x be a variable. In one variable, inequalities of the form ax + b > 0, ax + b < 0, ax + b?? 0 and ax + b < 0 are known as linear inequalities.

### Linear Inequation of Two Variables

Let x, y be variables, and a, b be non-zero real numbers. The inequation of the form ax + by < c, ax + by > c, ax + by ≤ c, and ax + by ≥ c are then referred to as linear inequities in the two variables x and y.

### Rules for Solving Inequalities

There are similar rules for solving inequalities to those for solving linear equations. However, when multiplying or dividing by a negative number, there is one exception. In order to solve inequalities, we can:

• Add the same number on both sides.

• From both sides, subtract the same number.

• By the same positive number, multiply both sides.

• By the same positive number, divide both sides.

• Multiply the same negative number on both sides and reverse the sign.

• Divide the same negative number between both sides and reverse the sign.

Ex: Solve x – 6 > 14

Solution:

x – 6 > 14

x – 6+ 6 > 14 + 6

x > 20

Ex. Solve the inequality 12 > 18 – y

Solution:

12 > 18 – y

18 – y < 12

18 – y – 18 < 12 –18

– y < –6

y > 6

### How to Represent the Solution of a Linear Inequality in One Variable on a Number Line

We use the following conventions to depict the solution of a linear inequality in one variable on a number line:

• If the inequality involves ‘≥’ or ‘≤’, we draw a filled circle(•) on the number line to indicate that the number corresponding to the filled circle is included in the solution set.

• If the inequality involves ‘>’ or ‘<’, we draw an open circle (O) on the number line to indicate that the number corresponding to the open circle is excluded from the solution set.

### How to Graphically Represent the Solution of a Linear Inequality

1. To represent the solution of a linear inequality in a plane graphically in one or two variables, we proceed as follows:

1. If the inequality involves ‘≥’ or ‘≤’, we draw the graph of the line as a thick line to indicate that the points on this line are included in the solution set.

2. If the inequality involves ‘>’ or ‘<’, we draw the graph of the line as a dotted line to indicate that the points on the line are excluded from the solution set.

2. Solution of a linear inequality in one variable can be represented on the number line as well as in the plane but the solution of a linear inequality in two variables of the type ax + by > c, ax + by ≥ c, ax + by < c or ax + by ≤ c (a ≠ 0, b ≠ 0) can be represented in the plane only.

3. A system of inequalities contains two or more inequalities taken together, and the solutions to the system of inequalities are the solutions common to all the inequalities that form the system.

### Important Results

1. If a, b ∈ R and b ≠ 0, then

(i) ab > 0 or a b > 0 ⇒ a and b are of the same sign.

(ii) ab < 0 or a b < 0 ⇒ a and b are of opposite sign.

1. If a is any positive real number, i.e., a > 0, then

(i) | x | < a ⇔ – a < x < a

| x | ≤ a ⇔ – a ≤ x ≤ a

(ii) | x | > a ⇔ x < – a or x > a

| x | ≥ a ⇔ x ≤ – a or x ≥ a

1. How Can One Solve Linear Inequalities?

Answer: Linear inequalities can be solved through:

1. The equation is rearranged in such a way that "y" is on the left, and everything else is on the right.

2. Plotting of the “y=” line. For y< or y>  make a dashed line and for y≤ or y≥ make a solid line.

3. In the case of above the line, one must shade for a “greater than” (y> or y≥) or below the line for a “less than” (y< or y≤).

2. What Symbols are Used in Linear Inequalities?

Answer: List of the symbol of inequality:

• < less than

• > greater than

• ≤ less than or equal to

• ≥ greater than or equal to

• ≠ not equal to

• = equal to SHARE TWEET SHARE SUBSCRIBE